Math problems

Question

Joleen and Pablo want to fertilize a rectangular garden with an area of A square feet. They know that 5 cups of fertilizer will cover an area of 240 square feet. They each write an equation to represent the relationship between the area, A, and the number of cups of fertilizer, c
J equation: A=48c
P equation: c=1/48A
Which of the equations is correct?

Answer 1

set the ratios of area to fertilizer equal to each other

the equations J and P are the same assuming P means c = (1/48)A and not c = 1/(48A)

math symbolic type would make this solution more clear

 

A/c = 240/5

c*(5/240)*(A/c) =(240/5)*(5/240)*c

(1/48) A = c

 

A/c = 240/5

A/c = 48

c*(A/c) = 48*c

A = 48c

 

A/c = 240/5
A/c = 240/5

Answer 2

There are two important concepts in this problem:

  1.  Which is the independent variable and which is the dependent variable?

  2.  Be sure that the units are used properly in any expression

 

“an equation to represent the relationship between the area, A, and the number of cups of fertilizer, c” means that we must find a function.  Either the area, A, is a function of the number of cups, c;  or else, the number of cups, c, is a function of the area, A.  Which is it?

 

A function has an independent variable which may change on its own (independently)  and when it does, the dependent variable (that depends on it) will change according to the function description.  Normally, we would ask, “How many cups of fertilizer would we need to fertilize this area A?”  So, c=f(A).

 

Now, a line has an inverse function (which is also a function).  So, we could describe A=f(c) by asking, “How large an area. A. will c cups of fertilizer cover?”

 

The relationship between A and c in this problem could be either of these, but the problem says, “an area of A square feet,” indicating that the size of the area is independent and that we are to find c knowing that 5 cops fertilizes 240 square feet.

 

The answer seems to be of the form:  c=f(A)

 

Let’s look at (2) and check the units.

 

We are given that 5 cups covers 240 square feet.  In math we write:

          (5 cups)/(240 square feet)

        5/240 cups per square foot

          1/48   cups per square foot

 

Now, when we multiply by the number of square feet, what happens?

       { (1 cup)/(48 square feet) )*(A square feet)

                   Square feet cancel out, leaving only cups

          (1/48)*A cups                      [ note: this is not 1/(48A) ]

 

Just to check this:

     What is   (48 square feet)/(1 cup) * (c cups) ?

        Cups cancel out, leaving only square feet, so we have the area A that c cups will fertilize.

 

Summary:

  Both formulas are o.k.. on units.  However, the formula  c=(1/48)A gives us the number of cups that Joleen and Pablo will need to “fertilize a rectangular garden with an area of A square feet.”

 

Let’s choose:    Pablo’s equation:    c=(1/48)A